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LESSON
Name
Date
4.1 Study Guide
For use with pages 171–176
GOAL
Write the prime factorization of a number.
VOCABULARY
EXAMPLE
Lesson 4.1
A prime number is a whole number that is greater than 1 and has
exactly two whole number factors, 1 and itself. A composite number
is a whole number that is greater than 1 and has more than two whole
number factors.
When you write a number as a product of prime numbers, you are
writing its prime factorization. You can use a diagram called a factor
tree to write the prime factorization of a number.
A monomial is a number, a variable, or the product of a number and
one or more variables raised to whole number powers.
1 Writing Factors
To play a game at a party, the guests need to divide into groups with more
than 1 person and no more than 10 people. There are 30 people at the party.
How many different group sizes are possible?
Solution
(1) Write 30 as a product of two whole numbers in all possible ways.
1 p 30
2 p 15
3 p 10
5p6
The factors of 30 are 1, 2, 3, 5, 6, 10, 15, and 30.
(2) Use the factors to find all the group sizes with more than 1 person but no
more than 10 people.
3 groups of 10
5 groups of 6
10 groups of 3
6 groups of 5
15 groups of 2
Answer: There are 5 possible group sizes.
Exercises for Example 1
Write all the factors of the number.
1. 42
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2. 19
3. 51
4. 21
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LESSON
Name
Date
4.1 Study Guide
Continued
EXAMPLE
For use with pages 171–176
2 Writing a Prime Factorization
Write the prime factorization of 540.
Lesson 4.1
One possible factor tree:
2
ⴢ
540
Write original number.
10 ⴢ 54
Write 540 as 10 p 54.
2
ⴢ
5
ⴢ
6
ⴢ
9
5
ⴢ
2
ⴢ
3
ⴢ
3
Write 10 as 2 p 5. Write 54 as 6 p 9.
ⴢ
3
Write 6 as 2 p 3. Write 9 as 3 p 3.
Another possible factor tree:
2
ⴢ
540
Write original number.
20 ⴢ 27
Write 540 as 20 p 27.
4
ⴢ
5
ⴢ
3
ⴢ
9
2
ⴢ
5
ⴢ
3
ⴢ
3
Write 20 as 4 p 5. Write 27 as 3 p 9.
ⴢ
3
Write 4 as 2 p 2. Write 9 as 3 p 3.
Both trees give the same result:
540 ⫽ 2 p 2 p 3 p 3 p 3 p 5 ⫽ 22 p 33 p 5.
Answer: The prime factorization of 540 is 22 p 33 p 5.
Exercises for Example 2
Tell whether the number is prime or composite. If it is composite, write its
prime factorization.
5. 243
EXAMPLE
6. 67
7. 78
8. 81
3 Factoring a Monomial
Factor the monomial 54x3y2.
54x3y 2 ⫽ 2 p 3 p 3 p 3 p x3 p y2
⫽2p3p3p3pxpxpxpypy
Write 54 as 2 p 3 p 3 p 3.
Write x3 as x p x p x and y2 as y p y.
Exercises for Example 3
Factor the monomial.
9. 16a4b
12
Pre-Algebra
Chapter 4 Resource Book
10. 25x5y2
11. 64r 3t 3
12. 19cd 6
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LESSON
Name
Date
4.2 Study Guide
For use with pages 177–181
GOAL
Find the GCF of two or more whole numbers.
VOCABULARY
A common factor is a whole number that is a factor of two or more
nonzero whole numbers. The greatest of the common factors is the
greatest common factor (GCF).
Two or more numbers are relatively prime if their greatest common
factor is 1.
EXAMPLE
1 Finding the Greatest Common Factor
A florist is making identical flower arrangements. There are 75 tulips, 60 irises,
30 hollyhocks, and 45 dahlias. What is the greatest number of arrangements that
can be made? How many tulips, irises, hollyhocks, and dahlias will be in each
arrangement?
Lesson 4.2
Method 1: List the factors of each number. Identify the greatest number that is
on every list.
Factors of 75:
Factors of 60:
Factors of 30:
Factors of 45:
1, 3, 5, 15 , 25, 75
1, 2, 3, 4, 5, 6, 10, 12, 15 , 20, 30, 60
1, 2, 3, 5, 6, 10, 15 , 30
1, 3, 5, 9, 15 , 45
The common factors
are 1, 3, 5, and 15.
The greatest common
factor is 15.
Method 2: Write the prime factorization of each number. The GCF is the
product of the common prime factors.
75 3 p 5 p 5
60 2 p 2 p 3 p 5
30 2 p 3 p 5
45 3 p 3 p 5
The common prime
factors are 3 and 5.
The GCF is the product
3 p 5 15.
Answer: The greatest number of flower arrangements than can be made is
15. Each arrangement will have 75 15 5 tulips, 60 15 4 irises,
30 15 2 hollyhocks, and 45 15 3 dahlias.
Exercises for Example 1
Find the greatest common factor of the numbers.
1. 27, 81
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Pre-Algebra
Chapter 4 Resource Book
2. 36, 48
3. 21, 49, 56
4. 45, 75, 90
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LESSON
Name
Date
4.2 Study Guide
Continued
EXAMPLE
For use with pages 177–181
2 Identifying Relatively Prime Numbers
Find the greatest common factor of the numbers. Then tell whether the numbers
are relatively prime.
a. 19, 57
b. 16, 81
Solution
a. List the factors of each number. Identify the greatest number that the lists
have in common.
Factors of 19: 1, 19
Factors of 57: 1, 3, 19, 57
The GCF is 19. So, the numbers are not relatively prime.
b. Write the prime factorization of each number.
16 24
81 34
There are no common prime factors. However, two numbers always have 1
as a common factor. So, the GCF is 1, and the numbers are relatively prime.
Lesson 4.2
Exercises for Example 2
Find the greatest common factor of the numbers. Then tell whether the
numbers are relatively prime.
5. 15, 99
EXAMPLE
6. 17, 85
7. 14, 27
8. 96, 204
3 Finding the GCF of Monomials
Find the greatest common factor of 121x 2y4 and 33x 3y2.
Factor the monomials. The GCF is the product of the common factors.
121x 2 y4 11 p 11 p x p x p y p y p y p y
33x 3y2 3 p 11 p x p x p x p y p y
Answer: The GCF is 11x 2 y2.
Exercises for Example 3
Find the greatest common factor of the monomials.
9. 300ab2, 125a2b
5 3
4
11. 75y z , 45y
Copyright © McDougal Littell/Houghton Mifflin Company
All rights reserved.
10. 12xy, 18x
12. 159x 5y5, 126xy
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LESSON
Name
Date
4.3 Study Guide
For use with pages 182–186
GOAL
Write equivalent fractions.
VOCABULARY
Two fractions that represent the same number are called equivalent
fractions.
A fraction is in simplest form when its numerator and its
denominator are relatively prime.
EXAMPLE
1 Writing Equivalent Fractions
18
27
Write two fractions that are equivalent to ᎏᎏ.
Multiply or divide the numerator and the denominator by the same nonzero
number.
18
18 p 2
36
ᎏᎏ ⫽ ᎏᎏ ⫽ ᎏᎏ
27
27 p 2
54
18
18 ⫼ 9
2
ᎏᎏ ⫽ ᎏᎏ ⫽ ᎏᎏ
27
27 ⫼ 9
3
Multiply numerator and denominator by 2.
Divide numerator and denominator by 9.
36
54
2
3
18
27
Answer: The fractions ᎏᎏ and ᎏᎏ are equivalent to ᎏᎏ.
Exercises for Example 1
Write two fractions that are equivalent to the given fraction.
8
10
Lesson 4.3
1. ᎏᎏ
EXAMPLE
5
45
2. ᎏᎏ
33
54
24
36
3. ᎏᎏ
4. ᎏᎏ
2 Writing a Fraction in Simplest Form
45
75
Write ᎏᎏ in simplest form.
Write the prime factorizations of the numerator and denominator.
45 ⫽ 32 p 5
75 ⫽ 3 p 52
The GCF of 45 and 75 is 3 p 5 ⫽ 15.
45
45 ⫼ 15
ᎏᎏ ⫽ ᎏᎏ
75
75 ⫼ 15
3
⫽ ᎏᎏ
5
30
Pre-Algebra
Chapter 4 Resource Book
Divide numerator and denominator by GCF.
Simplify.
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LESSON
Name
Date
4.3 Study Guide
Continued
EXAMPLE
For use with pages 182–186
3 Simplifying a Fraction
The average number of days with precipitation of 0.01 inch or more for
San Francisco, California, for the month of April is 6. Write the fraction, in
simplest form, of the days in April in San Francisco that have at least 0.01 inch
of precipitation, on average.
Solution
Number of days with at least 0.01 in. precipitation
6
ᎏᎏᎏᎏᎏᎏ ⫽ ᎏᎏ
30
Total number of days in April
6⫼6
⫽ ᎏᎏ
30 ⫼ 6
1
⫽ ᎏᎏ
5
Write fraction.
Divide numerator and
denominator by GCF, 6.
Simplify.
1
Answer: On average, ᎏᎏ of the days in April in San Francisco have at least
5
0.01 inch of precipitation.
Exercises for Examples 2 and 3
Write the fraction in simplest form.
8
12
20
25
5. ᎏᎏ
EXAMPLE
18
81
6. ᎏᎏ
27
36
7. ᎏᎏ
8. ᎏᎏ
4 Simplifying a Variable Expression
36x5y3
Write ᎏᎏ
4 in simplest form.
42x y
2
3
1
1
1
Factor numerator and denominator.
1
2冫2 p 冫
32 p x冫 p x p x p x p x p 冫y p y冫 p 冫y
⫽ ᎏᎏᎏᎏ
冫2 p 3冫 p 7 p 冫x p 冫y p 冫y p 冫y p y
Divide out common factors.
6x4
⫽ ᎏᎏ
7y
Simplify.
1
1
1
1
1
Lesson 4.3
36x5y3
22 p 32 p x p x p x p x p x p y p y p y
ᎏᎏ
ᎏᎏᎏᎏ
⫽
2p3p7pxpypypypy
42xy4
1
Exercises for Example 4
Write the fraction in simplest form.
24xy7
9. ᎏᎏ
2 3
60x y
Copyright © McDougal Littell/Houghton Mifflin Company
All rights reserved.
14x 2
49x y
10. ᎏᎏ
6
32x5y5
30xy
11. ᎏᎏ
2 3
12. ᎏᎏ2
64x y
35x
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LESSON
Name
Date
4.4 Study Guide
For use with pages 187–191
GOAL
Find the least common multiple of two numbers.
VOCABULARY
A multiple of a whole number is the product of the number and any
nonzero whole number. A multiple that is shared by two or more
numbers is a common multiple. The least of the common multiples
of two or more numbers is the least common multiple (LCM).
The least common denominator (LCD) of two or more fractions is
the least common multiple of the denominators.
EXAMPLE
1 Finding the Least Common Multiple
You have two alarm clocks set for the same time. The snooze time on one alarm
clock is 10 minutes and the snooze time on the other is 15 minutes. If you
continue hitting the snooze button on both alarm clocks after they ring the first
time together, in how many minutes will the alarm clocks ring together again?
Method 1: List the multiples of each number. Identify the least number that is
on both lists.
Multiples of 10: 10, 20, 30 , 40, 50, 60
Multiples of 15: 15, 30 , 45, 60, 75
The LCM of 10 and 15 is 30.
Method 2: Find the common factors of the numbers.
10 ⫽ 2 p 5
15 ⫽ 3 p 5
The common factor is 5.
Multiply all of the factors, using each common factor only once.
LCM ⫽ 2 p 3 p 5 ⫽ 30
Answer: Your alarm clocks will ring together again in 30 minutes.
EXAMPLE
2 Finding the Least Common Multiple of Monomials
Find the least common multiple of 20xy2 and 25xy3.
20xy2 ⫽ 2 p 2 p 5 p x p y p y
25xy3 ⫽ 5 p 5 p x p y p y p y
Common Factors are cirlcled
and used only once in the LCM.
LCM ⫽ 5 p x p y p y p 2 p 2 p 5 p y ⫽ 100xy3
Lesson 4.4
Answer: The least common multiple of 20xy2 and 25xy3 is 100xy3.
Exercises for Examples 1 and 2
Find the least common multiple of the numbers or monomials.
1. 5, 17
2. 15, 80
3. 45, 150
4. 24x10y4, 42x7y6
5. 13a3bc, 26ab2
6. 12x6y9, 16yz
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Chapter 4
Pre-Algebra
Resource Book
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LESSON
Name
Date
4.4 Study Guide
Continued
EXAMPLE
For use with pages 187–191
3 Comparing Fractions Using the LCD
Last year, a day spa had 48,000 visitors, including 18,000 men. This year, the day
spa had 54,000 visitors, including 24,000 men. In which year was the fraction of
men greater?
Solution
(1) Write the fractions and simplify.
18,000
48,000
3
8
24,000
54,000
Last year: ᎏᎏ ⫽ ᎏᎏ
3
8
4
9
This year: ᎏᎏ ⫽ ᎏᎏ
4
9
(2) Find the LCD of ᎏᎏ and ᎏᎏ.
The LCM of 8 and 9 is 72. So, the LCD of the fractions is 72.
(3) Write equivalent fractions using the LCD.
3
8
3p9
8p9
27
72
4
9
Last year: ᎏᎏ ⫽ ᎏᎏ ⫽ ᎏᎏ
27
72
4p8
9p8
32
72
This year: ᎏᎏ ⫽ ᎏᎏ ⫽ ᎏᎏ
32
72
3
8
4
9
(4) Compare the numerators: ᎏᎏ < ᎏᎏ, so ᎏᎏ < ᎏᎏ.
Answer: The fraction of men was greater this year.
EXAMPLE
4 Ordering Fractions and Mixed Numbers
7 24
12 9
13
6
Order the numbers 2 ᎏᎏ, ᎏᎏ, and ᎏᎏ from least to greatest.
7
12
2 p 12 ⫹ 7
12
31
12
(1) Write the mixed number as an improper fraction: 2 ᎏᎏ ⫽ ᎏᎏ ⫽ ᎏᎏ.
31 24
12 9
13
6
(2) Find the LCD of ᎏᎏ, ᎏᎏ, and ᎏᎏ.
The LCM of 12, 9, and 6 is 36. So, the LCD is 36.
(3) Write equivalent fractions using the LCD.
31
31 p 3
93
ᎏᎏ ⫽ ᎏᎏ ⫽ ᎏᎏ
12
12 p 3
36
24
24 p 4
96
ᎏᎏ ⫽ ᎏᎏ ⫽ ᎏᎏ
9
9p4
36
78
36
93
36
93
36
13
13 p 6
78
ᎏᎏ ⫽ ᎏᎏ ⫽ ᎏᎏ
6
6p6
36
96
36
13
6
7
12
7
12
24
9
(4) Compare the numerators: ᎏᎏ < ᎏᎏ and ᎏᎏ < ᎏᎏ, so ᎏᎏ < 2 ᎏᎏ and 2 ᎏᎏ < ᎏᎏ.
13
6
7
12
24
9
Answer: From least to greatest, the numbers are ᎏᎏ, 2 ᎏᎏ, and ᎏᎏ.
Lesson 4.4
Exercises for Examples 3 and 4
Order the numbers from least to greatest.
3 59 71
16 44 60
7. 1 ᎏᎏ, ᎏᎏ, ᎏᎏ
40
Pre-Algebra
Chapter 4 Resource Book
4 68
17 21
3
7
8. 3 ᎏᎏ, ᎏᎏ, 3 ᎏᎏ
1 41
12 6
2
13
9. 7 ᎏᎏ, ᎏᎏ, 7 ᎏᎏ
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LESSON
Name
Date
Lesson 4.5
4.5 Study Guide
For use with pages 193–198
GOAL
EXAMPLE
Multiply and divide powers.
1 Using the Product of Powers Property
The sun evaporates 1012 tons of water on Earth per day. Find the weight of water
evaporated by the sun in 105 days.
Solution
Tons of water
evaporated in 105 days
Tons of water
evaporated in one p Number of days
day
1012 p 105
Substitute values.
1012 5
Product of powers property
10
Add exponents.
17
Answer: The sun evaporates 1017 tons of water in 105 days.
Exercises for Example 1
Find the product. Write your answer using exponents.
1. 83 p 811
EXAMPLE
2. 7 p 76
3. 1119 p 114
4. 32 p 318 p 35
2 Using the Product of Powers Property
a. x6 p x8 x6 8
x14
Product of powers property
Add exponents.
b. 13x7 p 4x 4 13 p 4 p x7 p x 4
13 p 4 p x7 4
13 p 4 p x11
52x11
Commutative property of multiplication
Product of powers property
Add exponents.
Multiply.
Exercises for Example 2
Find the product. Write your answer using exponents.
5. x 3 p x12
48
Pre-Algebra
Chapter 4 Resource Book
6. c9 p c3
7. 4y p 3y9 p 4y5
8. 5k3 p 3k12 p 4k8
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LESSON
Name
Date
Continued
EXAMPLE
Lesson 4.5
4.5 Study Guide
For use with pages 193–198
3 Using the Quotient of Powers Property
96
9
a. 2 96 2
Quotient of powers property
94
12
17x
51x
Subtract exponents.
12 3
17x
51
Quotient of powers property
17x9
51
Subtract exponents.
x9
3
Divide numerator and denominator by 17.
b. 3 Exercises for Example 3
Find the quotient. Write your answer using exponents.
718
7
1521
15
9. 5
EXAMPLE
10. 13
x16
x
12k10
80k
11. 9
12. 7
4 Using Both Properties of Powers
18n8 p n3
81n
Simplify 9 .
18n8 p n3
18n8 3
9
81n
81n9
18n11
81n9
18n11 9
81
Product of powers property
Add exponents.
Quotient of powers property
18n2
81
Subtract exponents.
2n2
9
Divide numerator and denominator by 9.
Exercises for Example 4
Simplify.
a p a7
a
13. 5
Copyright © McDougal Littell/Houghton Mifflin Company
All rights reserved.
b7
b pb
14. 2 3
15x5 p x8
90x
15. 12
Chapter 4
4z13 p 5z18
25z p z
16. 5 12
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LESSON
Name
Date
4.6 Study Guide
For use with pages 199–203
GOAL
EXAMPLE
Work with negative and zero exponents.
1 Powers with Negative and Zero Exponents
Write the expression using only positive exponents.
1
6
Definition of negative exponent
b. x0h5 1 p h5
Definition of zero exponent
1
h
4y3
c. 4y3z7 z7
5
Lesson 4.6
a. 68 8
Definition of negative exponent
Definition of negative exponent
Exercises for Example 1
Write the expression using only positive exponents.
1. 71
EXAMPLE
3. 6x0y2
2. 1570
4. s11t1
2 Rewriting Fractions
Write the expression without using a fraction bar.
x8
y
1
27
a. b. 17
Solution
1
27
1
3
Write prime factorization of 27.
33
Definition of negative exponent
a. 3
x8
y
b. 17 x8y17
Definition of negative exponent
Exercises for Example 2
Write the expression without using a fraction bar.
1
64
5. Copyright © McDougal Littell/Houghton Mifflin Company
All rights reserved.
1
100
6. g5
c4
d
7. 2
8. 4
h
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Pre-Algebra
Resource Book
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LESSON
Name
Date
4.6 Study Guide
Continued
EXAMPLE
For use with pages 199–203
3 Using Powers Properties with Negative Exponents
Find the product or quotient. Write your answer using only positive exponents.
5x12
x
a. 1019 p 1023
b. 1
4
Lesson 4.6
Solution
a. 1019 p 1023 1019 (23)
104
1
10
4
5x12
x
12 14
b. 1
4 5x
Product of powers property
Add exponents.
Definition of negative exponent
Quotient of powers property
5x2
Subtract exponents.
5
x
2
Definition of negative exponent
Exercises for Example 3
Find the product or quotient. Write your answer using only positive exponents.
9. 49 p 417
EXAMPLE
10. 74 p 711
k12
k
11. 1
9
t15
t
12. 9
4 Solving Problems Involving Negative Exponents
The radius of a proton is about 1000 attometers. An attometer is 109 nanometer.
What is the radius of a proton in nanometers?
Solution
To find the radius of a proton in nanometers, multiply the radius of a proton in
attometers by the number of nanometers in one attometer.
1000 p 109 103 p 109
103 (9)
106
Rewrite 1000 as 103.
Product of powers property
Add exponents.
1
10
Definition of negative exponent
1
1,000,000
Evaluate power.
6
1
1,000,000
Answer: The radius of a proton is about nanometer.
58
Pre-Algebra
Chapter 4 Resource Book
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LESSON
Name
Date
4.7 Study Guide
For use with pages 204–209
GOAL
Write numbers using scientific notation.
VOCABULARY
A number is written in scientific notation if it has the form c ⫻ 10n
where 1 ≤ c < 10 and n is an integer.
EXAMPLE
1 Writing Numbers in Scientific Notation
a. Write 6000 in scientific notation.
Standard form
6000
Product form
6.0 ⫻ 1000
Scientific notation
6.0 ⫻ 103
Move decimal point
3 places to the left.
Exponent is 3.
Lesson 4.7
b. Write 0.0000017 in scientific notation.
Standard form
Product form
Scientific notation
0.0000017
1.7 ⫻ 0.000001
1.7 ⫻ 10⫺6
Exponent is ⴚ6.
Move decimal point
6 places to the right.
EXAMPLE
2 Writing Numbers in Standard Form
a. Write 1.15 ⫻ 105 in standard form.
Scientific notation
Product form
Standard form
1.15 ⫻ 105
1.15 ⫻ 100,000
115,000
Exponent is 5.
Move decimal point
5 places to the right.
b. Write 7.63 ⫻ 10⫺8 in standard form.
Scientific notation
Product form
Standard form
7.63 ⫻ 10⫺8
7.63 ⫻ 0.00000001
0.0000000763
Exponent is ⴚ8.
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Pre-Algebra
Chapter 4 Resource Book
Move decimal point
8 places to the left.
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LESSON
Name
Date
4.7 Study Guide
Continued
For use with pages 204–209
Exercises for Examples 1 and 2
Write the number in scientific notation.
1. 0.0000523
2. 81,200
3. 985,000,000
4. 0.0001289
7. 1.04 ⫻ 103
8. 1.24 ⫻ 106
Write the number in standard form.
5. 5.7 ⫻ 10⫺7
EXAMPLE
6. 3.524 ⫻ 10⫺5
3 Ordering Numbers Using Scientific Notation
Order 7.235 ⫻ 107, 8.6 ⫻ 106, and 8,900,000 from least to greatest.
(1) Write each number in scientific notation if necessary.
8,900,000 ⫽ 8.9 ⫻ 106
(2) Order the numbers with different powers of 10.
Because 106 < 107, 8.6 ⫻ 106 < 7.235 ⫻ 107 and 8.9 ⫻ 106 < 7.235 ⫻ 107.
(3) Order the numbers with the same power of 10.
Lesson 4.7
Because 8.6 < 8.9, 8.6 ⫻ 106 < 8.9 ⫻ 106.
(4) Write the original numbers in order from least to greatest.
8.6 ⫻ 106; 8,900,000; 7.235 ⫻ 107
Exercises for Example 3
Order the numbers from least to greatest.
9. 1.03 ⫻ 109; 2.8 ⫻ 107; 1,035,000,000
10. 9.25 ⫻ 10⫺5; 1.08 ⫻ 10⫺2; 0.013
EXAMPLE
4 Multiplying Numbers In Scientific Notation
Commutative and associative
properties of multiplication
(2.7 ⫻ 10⫺4)(7 ⫻ 10⫺8) ⫽ (2.7 ⫻ 7) ⫻ (10⫺4 ⫻ 10⫺8)
⫽ 18.9 ⫻ (10⫺4 ⫻ 10⫺8)
Multiply 2.7 and 7.
⫺4 ⫹ (⫺8)
⫽ 18.9 ⫻ 10
Product of powers property
⫺12
⫽ 18.9 ⫻ 10
Add exponents.
⫺11
⫽ 1.89 ⫻ 10
Write in scientific notation.
Exercises for Example 4
Find the product. Write your answer in scientific notation.
11. (1.2 ⫻ 105)(8.2 ⫻ 104)
Copyright © McDougal Littell/Houghton Mifflin Company
All rights reserved.
12. (7.2 ⫻ 10⫺2)(3.6 ⫻ 103)
Chapter 4
Pre-Algebra
Resource Book
67